Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682628 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3045 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18,22,25)(19,29,31,24)(20,33,32,44)(21,26,36,41)(23,28,39,27)(30,46,47,40)(34,42,35,48)(37,45,38,43);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,23)(21,35)(22,31)(25,29)(26,42)(27,33)(28,44)(30,37)(32,39)(34,36)(38,47)(40,45)(41,48)(43,46);(17,20,23,31,22,32,39,19)(18,24,27,33,25,29,28,44)(21,34,37,47,36,35,38,30)(26,40,43,42,41,46,45,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,25)(24,29)(26,41)(27,28)(33,44)(40,46)(42,48)(43,45);(3,11)(6,9)(10,16)(14,15)(17,21)(18,26)(19,30)(20,34)(22,36)(23,37)(24,40)(25,41)(27,43)(28,45)(29,46)(31,47)(32,35)(33,42)(38,39)(44,48)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591350);